Complex umbers D a (cos+ i ) = cos + i = cos + C cos (i) Ccos (i) (i) + + = cos + i cos si+ i cos + i = cos + i cos si cos i de Moivre s Theorem. Biomial expasio. cos + i si = cos + i cos si cos i Equatig the imagiary parts gives, si = cos si = ( ) si si = ( ) si = si si = si si Applyig cos =. si = si (as required) b (cos+ i) = cos+ isi = cos + C cos (isi ) + C cos (isi ) Ccos (i) C cos (i) (i) + + + = cos + i cos si+ i cos + i cos + i cossi + i si de Moivre s Theorem. Biomial expasio. cos + i si = cos + i cos si cos i cos + cossi + isi Equatig the imagiary parts gives, si = cos si cos + = (cos ) si cos + = ( ) si ( ) + = ( + ) ( ) + = si + + + = + si si = + (as required) Applyig cos = si. Pearso Educatio Ltd 8. Copyig permitted for purchasig istitutio oly. This material is ot copyright free.
c 7 (cos+ i) = cos7+ isi7 7 7 7 = cos + C cos (isi ) + C cos (isi ) 7 7 7 C cos (i) C cos (i) C cos (i) + + + 7 7 C cos (i) (i) + + 7 = cos + 7i cos si+ i cos + i cos si + i cos si 7 7 + 7i cossi + i si + i cos si de Moivre s Theorem. Biomial expasio. 7 cos 7+ i si 7 = cos + 7i cos si cos i cos + i cos + i cos si 7 7cossi isi Equatig the imagiary parts gives, 7 cos 7 = cos cos + cos 7 cos 7 = cos cos ( cos ) + cos ( cos ) 7 7 7 7cos ( cos ) = cos cos + cos + cos ( cos + cos ) = cos cos + cos 7 cos ( cos + cos cos ) 7 7 7 + cos 7 cos + cos 7 7 cos+ cos cos + 7 cos = cos cos + cos 7cos Applyig cos = si. 7 cos 7 = cos cos + cos 7 cos (as required) Pearso Educatio Ltd 8. Copyig permitted for purchasig istitutio oly. This material is ot copyright free.
d Let = cos+ isi + = (cos ) = cos C C C = + + + + = + + + + = + + + + = + + + + = cos + (cos ) + + = cos + =cos = + + So,cos cos 8cos = + + cos (cos cos ) cos = (cos + cos + ) Therefore, cos = (cos + cos + ) (as required) 8 e Let = cos + isi = (i si ) = i si = i si C C C C = + + + + + = + + + + + + + = + + = + = isi (isi ) + (i) = So, i = isi isi + i ( i) si = si si + si si = (si si + si ) = isi + =isi Therefore, = (si si + ) Pearso Educatio Ltd 8. Copyig permitted for purchasig istitutio oly. This material is ot copyright free.
a Let = cos + isi the we have cos = + We have ad ( cos isi) = + + = cos hece ( cos i cos si cos i cos cos i ) = + + + = ( cos isi) ( cos i cos si cos i cos cos i ) = + + Hece cos = ( cos cos + cos ) = cos cos + cossi = cos cos cos + cos ( cos ) = + + + cos cos cos cos ( cos cos ) = cos cos + cos b We have that cos = cos cos ad we wish to solve cos + cos = Usig the above idetities the equatio becomes cos cos + cos+ cos cos = Which simplifies to cos cos = Hece we either have cos = i which case = =.7or we have cos = Hece cos =± 8 Which implies we have =.7or =. Pearso Educatio Ltd 8. Copyig permitted for purchasig istitutio oly. This material is ot copyright free.
Let = cos + i si a b + = (cos ) = cos C C C C C = + + + + + + = + + + + + + = + + + + + = + + + + + + = cos + (cos ) + (si ) + + = cos = + + + So,cos cos cos cos = + + + cos cos cos cos (as required) cos d = (cos + cos + cos + )d si si si = + + + si si() si = + + + = + + + = + + 9 = + 9 = + 9 9 cos = + 9 9 a=, b= 9 () = cos Pearso Educatio Ltd 8. Copyig permitted for purchasig istitutio oly. This material is ot copyright free.
a We wish to show that cos = cos cos cos + Let us start with the right had side of the equatio, lettig = cos + isi we have cos = + = ( cos cos + cos ) = cos cos + cos si cos = ( + ) = ( cos cos + ) = cos cos + cos = cos Hece the right had side becomes cos cos cos + cos cos cos ( cos cos ) ( cos ) cos ( ) ( ) cos cos ( cos ) = + + + = + cos cos cos + + + = cos + cos cos cos cos + cos cos + + = cos cos cos + + = + + + cos si cos si cos si si = + + + cos si si si si si = cos si b We have cos d = cos cos cos + d = si si si + = + = 8 Pearso Educatio Ltd 8. Copyig permitted for purchasig istitutio oly. This material is ot copyright free.
si d a We wish to compute, let = cos + isi the we have si i = so that ( ) ( ) = = + + + i = ( cos cos + cos ) = ( cos + cos cos + ) Hece the itegral becomes ( cos cos cos ) d si si si + + = + + = b We wish to compute si cos d we have cos = = + From the previous part we kow that = cos + cos cos + ad ( ) ( ) = = i + + = + = + = ( cos) = ( cos ) ( cos 8cos ) ( cos cos ) Hece we have cos = + = cos cos + cos + So the itegral becomes cos cos + cos + d = si si + si + + = + + = + = = + 9 8 Pearso Educatio Ltd 8. Copyig permitted for purchasig istitutio oly. This material is ot copyright free. 7
c We wish to compute cos d Let = cos + isi the we have ( ) ( ) cos = + i i = + + + i = ( 8 + + + 8 ) i = ( i( si8 + si si si) ) = ( si8 + si si si ) 8 ( )( ) Hece the itegral becomes ( si8 si si si ) d 8 + = cos8 cos cos cos 8 + + 8 79 7 = 8 + + + + = = 8 8 8 8 a Let = cos + isi the we have cos = ( + ) = ( cos+ i si) + ( cos i si) Notig that odd powers will cacel this simplifies to = ( cos cos si + cos si si ) = cos cos + cos = cos cos cos + cos cos cos cos 8cos + 8cos Pearso Educatio Ltd 8. Copyig permitted for purchasig istitutio oly. This material is ot copyright free. 8
b We wish to solve x 8x + 8x = We use the substitutio x= cos so that the equatio becomes cos 8cos + 8cos = i.e. cos 8cos + 8cos = Hece cos = The geeral solutio to this is give by =± + k where k is ay iteger i.e. =± + k 8 Tryig both choices of sig ad varyig k gives the followig values of x=±.98 x=±. x= cos x=±. But these are distict solutios ad sice the polyomial has order there are at most uique solutios hece these are all solutios. 7 a (cos + i) = cos + isi = cos + C cos (i) + C cos (i) + C cos (i) + (i) = cos + i cos si + i cos + i cossi + i de Moivre s Theorem. Biomial expasio. = cos + i cos si cos i cos + cos + i si = cos + i cos si cos i cos + () Equatig the imagiary parts of () gives: si = cos si cos (as required) Pearso Educatio Ltd 8. Copyig permitted for purchasig istitutio oly. This material is ot copyright free. 9
7 b Equatig the real parts of () gives: = + cos cos cos si si ta = = si cos si cos si (si cos ) cos cos cos si + si (cos cos ) cossi cossi = cos cos cos cos si si cos cos + cos cos si cos = cos cos si cos cos cos cossi + cos coscos cos = ta ta ta + ta Therefore, ta ta ta = ta + ta (as required) c x x x x + + = x x + = x x x x = () x x + Let x= ta ; the () ta ta = ta ta + ta = α = From part b. 9 =,,,, 9 =,,,, 9 x= ta = ta,ta,ta,ta x=.989,.9,.7,.87, x=.,,,.,.7 ( d.p.) Pearso Educatio Ltd 8. Copyig permitted for purchasig istitutio oly. This material is ot copyright free.